quawei {lmomco} | R Documentation

## Quantile Function of the Weibull Distribution

### Description

This function computes the quantiles of the Weibull distribution given parameters (zeta, β, and delta) of the distribution computed by `parwei`. The quantile function of the distribution is

x(F) = β[- log(1-F)]^{1/delta} - zeta mbox{,}

where x(F) is the quantile for nonexceedance probability F, zeta is a location parameter, β is a scale parameter, and delta is a shape parameter.

The Weibull distribution is a reverse Generalized Extreme Value distribution. As result, the Generalized Extreme Value algorithms are used for implementation of the Weibull in this package. The relation between the Generalized Extreme Value distribution parameters (xi, α, and kappa) is

kappa = 1/delta mbox{,}

α = β/delta mbox{, and}

xi = zeta - β mbox{.}

These relations are taken from Hosking and Wallis (1997).

In R the quantile function of the Weibull distribution is `qweibull`. Given a Weibull parameter object `p`, the R syntax is `qweibull(f, p$para[3], scale=p$para[2]) - p$para[1]`. For the current implementation for this package, the reversed Generalized Extreme Value distribution is used `-quagev((1-f),para)`.

### Usage

    quawei(f, para)

### Arguments

`f` | Nonexceedance probability (0 <= F <= 1).
`para` | The parameters from `parwei` or similar.

### Value

Quantile value for nonexceedance probability F.

### Author(s)

W.H. Asquith

### References

Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

### See Also

`cdfwei`, `parwei`

### Examples

      # Evaluate Weibull deployed here and within R (qweibull)
      lmr <- lmom.ub(c(123,34,4,654,37,78))
      WEI <- parwei(lmr)
      Q1  <- quawei(0.5,WEI)
      Q2  <- qweibull(0.5,shape=WEI$para[3],scale=WEI$para[2])-WEI$para[1]
      if(Q1 == Q2) EQUAL <- TRUE

      # The Weibull is a reversed generalized extreme value
      Q <- sort(rlmomco(34,WEI)) # generate Weibull sample
      lm1 <- lmoms(Q)    # regular L-moments
      lm2 <- lmoms(-Q)   # L-moment of negated (reversed) data
      WEI <- parwei(lm1) # parameters of Weibull
      GEV <- pargev(lm2) # parameters of GEV
      F <- nonexceeds()  # Get a vector of nonexceedance probs
      plot(pp(Q),Q)
      lines(F,quawei(F,WEI))
      lines(F,-quagev(1-F,GEV),col=2) # line over laps previous

[Package lmomco version 0.92 Index]